\documentclass[reqno]{amsart}
\usepackage{hyperref}
\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 209, pp. 1--9.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2013 Texas State University - San Marcos.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE-2013/209\hfil Boundedness in a chemotaxis system]
{Boundedness in a chemotaxis system with consumption of
chemoattractant and \\ logistic source}
\author[L. Wang, S. U.-D. Khan, S. U.-D. Khan \hfil EJDE-2013/209\hfilneg]
{Liangchen Wang, Shahab Ud-Din Khan, Salah Ud-Din Khan}
\address{Liangchen Wang \newline
College of Mathematics and Statistics,
Chongqing University, Chongqing 401331, China}
\email{liangchenwang324@126.com}
\address{Shahab Ud-Din Khan \newline
College of Mathematics and Statistics,
Chongqing University, Chongqing 401331, China}
\email{sudkhan@163.com}
\address{Salah Ud-Din Khan \newline
College of Engineering, King Saud University, P.O. Box 800,
Riyadh 11421, Kingdom of Saudi Arabia}
\email{salahudkhan@126.com}
\thanks{Submitted July 1, 2013. Published September 19, 2013.}
\subjclass[2000]{35B35, 35K55, 92C17}
\keywords{Chemotaxis; global existence; boundedness; logistic source}
\begin{abstract}
In this article, we consider a chemotaxis system with consumption
of chemoattractant and logistic source
\begin{gather*}
u_t=\Delta u-\chi\nabla\cdot(u\nabla v)+f(u),\quad x\in \Omega,\; t>0,\\
v_t=\Delta v-uv,\quad x\in\Omega,\; t>0,
\end{gather*}
under homogeneous Neumann boundary conditions in a smooth bounded domain
$\Omega\subset \mathbb{R}^n$, with non-negative initial data $u_0$ and
$v_0$ satisfying $(u_0,v_0)\in (W^{1,\theta}{(\Omega)})^2$
(for some $\theta>n$). $\chi>0$ is a parameter referred to as
chemosensitivity and $f(s)$ is assumed to generalize the logistic function
\[
f(s)=as-bs^2,\quad s\geq0,\text{ with } a>0,\;b>0.
\]
It is proved that if $\|v_0\|_{L^\infty(\Omega)}>0$ is sufficiently small
then the corresponding initial-boundary value problem possesses a unique
global classical solution that is uniformly bounded.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks
\section{Introduction}
This article considers the following chemotaxis system with consumption
of chemoattractant and logistic source
\begin{equation} \label{e1.1}
\begin{gathered}
u_t=\Delta u-\chi\nabla\cdot(u\nabla v)+f(u),\quad x\in \Omega,\; t>0,\\
v_t=\Delta v-uv,\quad x\in \Omega,\; t>0,\\
\frac{\partial u}{\partial \nu}=\frac{\partial v}{\partial \nu}=0,\quad
x\in \partial\Omega,\; t>0,\\
u(x,0)=u_0(x),\quad v(x,0)=v_0(x),\quad x\in \Omega,
\end{gathered}
\end{equation}
where $\Omega\subset \mathbb{R}^n$ is a bounded domain with smooth
boundary $\partial\Omega$, and $\partial/\partial\nu$ denotes the
derivative with respect to the outer normal of $\partial\Omega$.
The parameter $\chi>0$ is referred as chemosensitivity, and the
function $f\in C^1([0,\infty))$ with $f(0)=0$. Moreover, we shall
suppose that
\begin{equation} \label{e1.2}
f(u)\leq au-bu^p\quad \text{for all } u\geq0
\end{equation}
with some $a>0$, $b>0$ and $p>1$.
Equations \eqref{e1.1} is the well-known Keller-Segel model, and the
origin of this fundamental model was introduced by Keller and Segel \cite{1}
to describe the motion of cells which are diffusing and moving towards
the concentration gradient of a chemical signal substance called
chemoattractant, the latter being produced by the cells themselves.
We refer the reader to the paper \cite{2} where a comprehensive
information of further examples illustrating the outstanding biological
relevance of chemotaxis can be found. In this paper, we consider
a mathematical model for the motion of cells which towards the higher
concentration of oxygen that is consumed by the cells, where $u=u(x, t)$
denotes the density of the cells and $v=v(x, t)$ represents the concentration
of the oxygen.
During the past four decades, the Keller-Segel models have become one of
the best-study models in mathematical biology the Keller-Segel models
have been studied extensively by many authors. For example, Keller
and Segel \cite{1} proposed the following classical chemotaxis model
\begin{equation} \label{e1.3}
\begin{gathered}
u_t=\Delta u-\nabla\cdot(u\nabla v),\quad x\in \Omega,\; t>0,\\
v_t=\Delta v-v+u,\quad x\in\Omega,\; t>0,
\end{gathered}
\end{equation}
which has been investigated successfully up to now and the main issue
of the investigation was the solutions of the model are bounded or blow-up.
If $n=1$, in \cite{3}, it was shown all solutions of \eqref{e1.3} are
global in time and bounded; if $n=2$, then all solutions of \eqref{e1.3}
are global in time and bounded provided that $\|u_0\|_{L^1{(\Omega)}}<4\pi$
in \cite{4}, however, for almost every $\|u_0\|_{L^1{(\Omega)}}>4\pi$,
then the corresponding solutions of \eqref{e1.3} blow up either in finite
or infinite time in \cite{5} and that some radially symmetric solutions
blow up in finite time in \cite{6,7}; if $n\geq3$,
Winkler \cite{8} showed that $\|u_0\|_{L^{n/2+\epsilon}{(\Omega)}}$
and $\|\nabla v_0\|_{L^{n+\epsilon}{(\Omega)}}$ are small for all
$\epsilon>0$, then the solution is global in time and bounded, however,
for any $\|u_0\|_{L^1{(\Omega)}}>0$, then the radially symmetric
solution of \eqref{e1.3} blows up either in finite or infinite time
(see also \cite{7,9,10}).
Involving a source term of logistic type in chemotaxis system have
been studied \cite{2,11,12,13,14,h2}. The following initial-boundary
value chemotaxis model with logistic source
\begin{equation} \label{e1.4}
\begin{gathered}
u_t=\Delta u-\nabla\cdot(u\chi(v)\nabla v)+f(u),\quad x\in \Omega,\; t>0,\\
v_t=\Delta v-v+u,\quad x\in\Omega,\; t>0.
\end{gathered}
\end{equation}
If $\chi(v)$ is a constant, Winkler \cite{11} studied proved that the
solutions of problem \eqref{e1.4} are global and bounded provided
that $f(0)\geq0$ as well as $f(u)\leq a-bu^2$ with some $a\geq0$ and $b$
is sufficiently large. If
$\chi(v)\leq\frac{\chi_0}{(1+\beta v)^\delta}$ for all $v\geq0$
and some $\delta>1$, $\chi_0>0$ and $\beta>0$, the authors \cite{h2}
shown that the model \eqref{e1.4} with logistic source $f(u)$
satisfies \eqref{e1.2} with $p=2$ then solutions are global and bounded
provided that $\chi_0$ and $a$ are sufficiently small,
the authors \cite{hh} recent obtain the same result for all positive
values of $\chi_0$ and $a$, which improved the previous result.
The model \eqref{e1.1} deals with the chemotaxis process where the
signal is consumed by the cells, rather than produced by the cells.
In the absence of the logistic source (i.e. $f(u)\equiv0$) for
problem \eqref{e1.1}, Tao \cite {18} proved that the classical solution
of model \eqref{e1.1} is uniformly bounded provided that
$\|v_0\|_{L^\infty(\Omega)}$ is sufficiently small.
In particular, if $\Omega\subset \mathbb{R}^3$ is a bounded convex domains,
Tao and Winkler \cite {27} showed that there exists $T>0$ such that
the problem has global weak solution which is bounded and smooth in
$\Omega\times(T,+\infty)$. It is the goal of this paper to prove
that model \eqref{e1.1} has global and bounded solutions provided that
$\|v_0\|_{L^\infty(\Omega)}>0$ is sufficiently small (Theorem \ref{thm3.2}).
\section{Preliminaries}
We first state one result concerning local-in-time existence of a
classical solution to problem \eqref{e1.1}.
\begin{theorem} \label{thm2.1}
Let the non-negative functions $u_0$ and $v_0$ satisfy
$(u_0,v_0)$ belong to $(W^{1,\theta}{(\Omega)})^2$,
for some $\theta>n$.
Moreover, $f(s)$ with $s\geq0$ is smooth and $f(0)=0$.
Then problem \eqref{e1.1} has a unique local-in-time non-negative
classical solution
\begin{equation} \label{e2.1}
(u, v)\in (C([0,T_{\rm max}); W^{1,\theta}{(\Omega)})
\cap C^{2,1}(\overline{\Omega}\times(0,T_{\rm max})))^2,
\end{equation}
where $T_{\rm max}$ denotes the maximal existence time.
If for every $T0$ such that $u(x,t)$ and $v(x,t)$ satisfy
\begin{gather} \label{e2.3}
\|u(\cdot,t)\|_{L^1(\Omega)}\leq C_0, \\
\label{e2.4}
0\leq v\leq \|v_0\|_{L^\infty(\Omega)}
\end{gather}
for all $t\in(0,T_{\rm max})$.
\end{theorem}
\begin{proof}
As in \cite{21,18,24}, let $V=(u,v)\in \mathbb{R}^2$. Then the
initial-boundary value problem \eqref{e1.1} can be reformulated as
\begin{gather*}
V_t=\nabla\cdot(F(V)\nabla V)+H(V)\\
\frac{\partial V}{\partial \nu}=0,\quad x\in \partial\Omega,\; t>0,\\
V(x,0)=(u_0(x),v_0(x)),\quad x\in \Omega,
\end{gather*}
where
\[
F(V)= \begin{pmatrix}
1 & -\chi u \\
0 & 1 \\
\end{pmatrix}, \quad
H(V)=\begin{pmatrix} f(u) \\ -uv\end{pmatrix}.
\]
Then applying \cite[Theorems 14.4, 14.6, 15.5]{d4},
statements \eqref{e2.1} and \eqref{e2.2} can be proved.
Since the initial data $u_0\geq0$, $v_0\geq0$ and $f(0)=0$,
the maximum principle ensures that both $u$ and $v$ are non-negative.
By the maximum principle we have \eqref{e2.4}.
Now, we prove \eqref{e2.3}. Integrating the first equation in \eqref{e1.1}
and using \eqref{e1.2}, we obtain
\begin{equation} \label{e2.5}
\frac{d}{dt}\int_{\Omega}udx=\int_{\Omega}f(u)dx\leq\int_{\Omega}au-bu^pdx.
\end{equation}
By Young's inequality, since $b>0$ and $p>1$, we obtain
\begin{equation} \label{e2.6}
(a+1)\int_{\Omega}u\,dx
=\int_{\Omega}b^{-1/p}(a+1)b^{1/p}u\,dx
\leq \int_{\Omega}bu^pdx+(a+1)^\frac{p}{p-1}b^\frac{1}{1-p}|\Omega|.
\end{equation}
Combining \eqref{e2.5} and \eqref{e2.6}, we conclude
\begin{equation} \label{e2.7}
\frac{d}{dt}\int_{\Omega}udx+\int_{\Omega}u\,dx
\leq (a+1)^\frac{p}{p-1}b^\frac{1}{1-p}|\Omega|.
\end{equation}
Integrating, we have
\[
\int_{\Omega}udx\leq c_0, \quad
c_0=\max \{\|u_0\|_{L^1(\Omega)}, (a+1)^\frac{p}{p-1}b^\frac{1}{1-p}|\Omega|\}>0.
\]
\end{proof}
Let us collect some basic statements about the Gagliardo-Nirenberg
inequality which will be used in forthcoming proofs. For details,
we refer the reader to \cite{28,29,30} (see also \cite{9,10}).\
\begin{lemma} \label{lem2.2}
Let
\[
\alpha^*=\begin{cases}
\frac{2n}{n-2}, &\text{if }n>2,\\
\infty, &\text{if }n=1,2.
\end{cases}
\]
Then for all $l^*\in(2,\infty)$ satisfying $l^*\leq \alpha^*$
and $h\in(0,2)$, $\alpha\in[h,l^*]$, there exists a constant $c_{GN}>0$
such that
\[
\|\psi\|_{L^\alpha(\Omega)}\leq c_{GN}(\|\nabla \psi\|_{L^2(\Omega)}^{\lambda^*}
\|\psi\|_{L^h(\Omega)}^{1-\lambda^*}+\|\psi\|_{L^h(\Omega)})
\]
holds for any $\psi\in W^{1,2}(\Omega)$, where
$\lambda^*=\frac{\frac{n}{h}-\frac{n}{\alpha}}{1-\frac{n}{2}+\frac{n}{h}}$.
\end{lemma}
\section{Global bounded solutions}
The main step towards the existence and boundedness of a global solution
is to establish uniform bound of the cells population density $u(x,t)$ in
the space $L^{n+1}(\Omega)$. This is accomplished by providing some
associated weighted bounds involving weight functions $\phi(v)$
which are uniformly bounded both from above and below by positive constants.
This approach was developed by Winkler in \cite{16} (see also \cite{h2,18}).
\begin{lemma} \label{lem3.1}
Let $f(u)$ satisfy \eqref{e1.2}, $\|v_0\|_{L^\infty(\Omega)}>0$ and $\chi>0$.
Then there exists a constant $C>0$ such that the first component of
the solution of \eqref{e1.1} satisfies
\begin{equation} \label{e3.1}
\|u(\cdot,t)\|_{L^{n+1}(\Omega)}\leq C\quad \text{for all }
t\in(0,T_{\rm max}).
\end{equation}
\end{lemma}
\begin{proof}
Set $k:=n+1$ and fix $\|v_0\|_{L^\infty(\Omega)}>0$ small such that
\begin{equation} \label{e3.2}
\|v_0\|_{L^\infty(\Omega)}\leq\frac{1}{6(n+1)\chi}.
\end{equation}
Define
$$
\phi(s):=e^{(\alpha s)^2}\quad \text{for all }
0\leq s\leq \|v_0\|_{L^\infty(\Omega)},
$$
where
\[
\alpha=\sqrt{\frac{n}{24(n+1)}}\frac{1}{\|v_0\|_{L^\infty(\Omega)}}.
\]
By direct calculation, from \eqref{e1.1}, we obtain
\begin{align}
&\frac{1}{k}\frac{d}{dt}\int_{\Omega}u^k \phi(v)dx \nonumber \\
&=\int_{\Omega}u^{k-1}\phi(v)u_t dx+\frac{1}{k}\int_{\Omega}u^k \phi'(v)v_t\,dx
\nonumber\\
&=\int_{\Omega}u^{k-1}\phi(v)\Delta u dx
-\int_{\Omega}u^{k-1}\phi(v)\chi\nabla\cdot(u\nabla v)dx
+\int_{\Omega}u^{k-1}\phi(v)f(u)dx \nonumber \\
&\quad +\frac{1}{k}\int_{\Omega}u^k \phi'(v)\Delta v\,dx
-\frac{1}{k}\int_{\Omega}u^{k+1}v \phi'(v)dx \nonumber \\
&=-(k-1)\int_{\Omega}u^{k-2}\phi(v)|\nabla u|^2dx
-\int_{\Omega}u^{k-1} \phi'(v)\nabla u\cdot\nabla v\,dx \nonumber \\
&\quad+\chi(k-1)\int_{\Omega}u^{k-1}\phi(v)\nabla u\cdot\nabla v\,dx
+\chi\int_{\Omega}u^k\phi'(v)|\nabla v|^2dx \nonumber \\
&\quad +\int_{\Omega}u^{k-1}\phi(v)f(u)dx-\int_{\Omega}u^{k-1}
\phi'(v)\nabla u\cdot\nabla v\,dx \nonumber \\
&\quad -\frac{1}{k}\int_{\Omega}u^k \phi''(v)|\nabla v|^2dx
-\frac{1}{k}\int_{\Omega}u^{k+1}v \phi'(v)dx. \label{e3.3}
\end{align}
Since $f(s)\leq as-bs^p$ and $\phi'(s)\geq0$ for all $s\geq0$, we have
\begin{equation} \label{e3.4}
\begin{split}
&\frac{1}{k}\frac{d}{dt}\int_{\Omega}u^k \phi(v)dx
+(k-1) \int_{\Omega}u^{k-2}\phi(v)|\nabla u|^2dx
+\frac{1}{k} \int_{\Omega}u^k \phi''(v)|\nabla v|^2dx\\
&\leq -2\int_{\Omega}u^{k-1} \phi'(v)\nabla u\cdot\nabla v\,dx
+\chi(k-1)\int_{\Omega}u^{k-1}\phi(v)\nabla u\cdot\nabla v\,dx\\
&\quad +\chi\int_{\Omega}u^k\phi'(v)|\nabla v|^2dx
+a\int_{\Omega}u^k\phi(v)dx-b\int_{\Omega}u^{k+p-1}\phi(v)dx.
\end{split}
\end{equation}
By Young's inequality, we obtain
\begin{equation} \label{e3.5}
\begin{split}
-2\int_{\Omega}u^{k-1} \phi'(v)\nabla u\cdot\nabla v\,dx
&\leq\frac{k-1}{4}\int_{\Omega}u^{k-2}\phi(v)|\nabla u|^2dx\\
&+\frac{4}{k-1}\int_{\Omega}u^k\frac{\phi'^2(v)}{\phi(v)}|\nabla v|^2dx
\end{split}
\end{equation}
and
\begin{equation} \label{e3.6}
\begin{split}
\chi(k-1)\int_{\Omega}u^{k-1}\phi(v)\nabla u\cdot\nabla v\,dx
&\leq\frac{k-1}{4}\int_{\Omega}u^{k-2}\phi(v)|\nabla u|^2dx\\
& +\chi^2(k-1)\int_{\Omega}u^k\phi(v)|\nabla v|^2dx.
\end{split}
\end{equation}
Thus, from \eqref{e3.4}--\eqref{e3.6} we obtain
\begin{equation} \label{e3.7}
\begin{split}
&\frac{1}{k}\frac{d}{dt}\int_{\Omega}u^k \phi(v)dx
+\frac{k-1}{2}\int_{\Omega}u^{k-2}\phi(v)|\nabla u|^2dx
+\frac{1}{k}\int_{\Omega}u^k \phi''(v)|\nabla v|^2dx\\
&\leq \frac{4}{k-1}\int_{\Omega}u^k\frac{\phi'^2(v)}{\phi(v)}|\nabla v|^2dx
+\chi^2(k-1)\int_{\Omega}u^k\phi(v)|\nabla v|^2dx\\
&\quad +\chi\int_{\Omega}u^k\phi'(v)|\nabla v|^2dx
+a\int_{\Omega}u^k\phi(v)dx-b\int_{\Omega}u^{k+p-1}\phi(v)dx.
\end{split}
\end{equation}
Next we show that the three terms on the right-hand side of \eqref{e3.7}
are dominated by $\frac{1}{k}\int_{\Omega}u^k \phi''(v)|\nabla v|^2dx$.
To this end, for $s\geq0$, we compute
\begin{gather*}
y_1(s):=\frac{\phi''(s)}{k}=\frac{2}{k}\alpha^2e^{(\alpha s)^2}+\frac{4}{k}\alpha^4s^2e^{(\alpha s)^2},\\
y_2(s):=\frac{4}{k-1}\frac{\phi'^2(s)}{\phi(s)}=\frac{16}{k-1}\alpha^4s^2e^{(\alpha s)^2},\\
y_3(s):=\chi^2(k-1)\phi(s)=\chi^2(k-1)e^{(\alpha s)^2},\\
y_4(s):=\chi\phi'(s)=2\chi\alpha^2se^{(\alpha s)^2}.
\end{gather*}
By a direct calculation, we obtain
\begin{equation} \label{e3.8}
\frac{y_2(s)}{\frac{1}{3}y_1(s)}
\leq \frac{\frac{16}{k-1}\alpha^4s^2e^{(\alpha s)^2}}{\frac{2}{3k}
\alpha^2e^{(\alpha s)^2}}
= \frac{24k}{k-1}(\alpha s)^2
\leq \frac{24(n+1)}{n}(\alpha \|v_0\|_{L^\infty(\Omega)})^2
=1,
\end{equation}
where we have used that
$\alpha=\sqrt{\frac{n}{24(n+1)}}\frac{1}{\|v_0\|_{L^\infty(\Omega)}}$.
Using \eqref{e3.2},
\begin{equation} \label{e3.9}
\frac{y_3(s)}{\frac{1}{3}y_1(s)}
\leq \frac{\chi^2(k-1)e^{(\alpha s)^2}}{\frac{2}{3k}\alpha^2e^{(\alpha s)^2}}
\leq \frac{3k(k-1)\chi^2}{2\alpha^2}
=36(n+1)^2\|v_0\|^2_{L^\infty(\Omega)}\chi^2
\leq 1
\end{equation}
and
\begin{equation} \label{e3.10}
\frac{y_4(s)}{\frac{1}{3}y_1(s)}
\leq \frac{2\chi\alpha^2se^{(\alpha s)^2}}{\frac{2}{3k}\alpha^2e^{(\alpha s)^2}}
\leq 3k\chi s
\leq 3(n+1)\chi\|v_0\|_{L^\infty(\Omega)}
\leq \frac{1}{2}.
\end{equation}
Therefore, from \eqref{e3.7}-\eqref{e3.10}, it follows easily that
\begin{equation} \label{e3.11}
\begin{split}
&\frac{d}{dt}\int_{\Omega}u^k \phi(v)dx
+kb\int_{\Omega}u^{k+p-1}\phi(v)dx
+\frac{2(k-1)}{k}\int_{\Omega}|\nabla u^{k/2}|^2\phi(v)dx\\
&\leq ka\int_{\Omega}u^k\phi(v)dx.
\end{split}
\end{equation}
Since $ 0\leq s\leq \|v_0\|_{L^\infty(\Omega)}$, we have
$1\leq\phi(s)\leq e^{(\alpha\|v_0\|_{L^\infty(\Omega)})^2}:=d$,
it is not difficult to obtain
\begin{equation} \label{e3.12}
kb\int_{\Omega}u^k\phi(v)dx\leq kb\int_{\Omega}u^{k+p-1}\phi(v)dx+kbd|\Omega|.
\end{equation}
Combining \eqref{e3.11} with \eqref{e3.12} yields
\begin{equation} \label{e3.13}
\begin{split}
&\frac{d}{dt}\int_{\Omega}u^k \phi(v)dx
+kb\int_{\Omega}u^k\phi(v)dx
+\frac{2(k-1)}{k}\int_{\Omega}|\nabla u^{k/2}|^2\phi(v)dx\\
&\leq ka\int_{\Omega}u^k\phi(v)dx+kbd|\Omega|.
\end{split}
\end{equation}
Using Lemma \ref{lem2.2} and $(x+y)^\gamma\leq2^\gamma(x^\gamma+y^\gamma)$
for all $x,y\geq0$ and $\gamma>0$, we obtain
\begin{equation} \label{e3.14}
\begin{split}
ka\int_{\Omega}u^k\phi(v)dx
&\leq kad\int_{\Omega}u^kdx\\
&=kad\|u^{k/2}\|_{L^2(\Omega)}^2\\
&\leq kad(c_{GN}\|\nabla u^{k/2}\|_{L^2(\Omega)}^\lambda
\|u^{k/2}\|_{L^{2/k}(\Omega)}^{1-\lambda}
+c_{GN}\|u^{k/2}\|_{L^{2/k}(\Omega)})^2\\
&\leq 4kad(c^2_{GN}c_0^{k(1-\lambda)}
\|\nabla u^{k/2}\|_{L^2(\Omega)}^{2\lambda}+c^2_{GN}c_0^k)
\end{split}
\end{equation}
holds with some constant $c_{GN}>0$ and
$$
\lambda=\frac{\frac{kn}{2}-\frac{n}{2}}{1-\frac{n}{2}+\frac{kn}{2}}\in(0,1).
$$
By Young's inequality, we derive
\begin{equation} \label{e3.15}
\begin{split}
ka\int_{\Omega}u^k\phi(v)dx
&\leq 4kadc^2_{GN}c_0^{k(1-\lambda)}\|\nabla u^{k/2}\|_{L^2(\Omega)}
^{2\lambda}+4kadc^2_{GN}c_0^k\\
&\leq \frac{2(k-1)}{k}\int_{\Omega}|\nabla u^{k/2}|^2dx+c_1\\
&\leq \frac{2(k-1)}{k}\int_{\Omega}|\nabla u^{k/2}|^2\phi(v)dx+c_1,
\end{split}
\end{equation}
where
\[
c_1=c_0^k(4kadc^2_{GN}(\frac{2k-2}{k})^{-\lambda})^{\frac{1}{1-\lambda}}
+4kadc^2_{GN}c_0^k>0.
\]
Hence, substituting \eqref{e3.15} into \eqref{e3.13} yields
\begin{equation} \label{e3.16}
\frac{d}{dt}\int_{\Omega}u^k \phi(v)dx+kb\int_{\Omega}u^k\phi(v)dx
\leq c_1+kbd|\Omega|.
\end{equation}
Integrating \eqref{e3.16}, we have
\begin{equation*}
\int_{\Omega}u^kdx\leq\int_{\Omega}u^k \phi(v)dx
\leq \max \big\{d\int_{\Omega}u_0^k, \frac{c_1+kbd|\Omega|}{kb}\big\},
\end{equation*}
we arrive at the desired result.
\end{proof}
\begin{remark} \label{rmk1} \rm
To prove that the three terms on the right-hand side of \eqref{e3.7}
are dominated by $\frac{1}{k}\int_{\Omega}u^k \phi''(v)|\nabla v|^2dx$,
we need $\frac{y_i(s)}{\frac{1}{3}y_1(s)}\leq1$ $(i=2,3,4)$, so we have
$0n$.
So to avoid this situation, we choose $k:=n+1$ in Lemma \ref{lem3.1}.
\end{remark}
We are now in a position to prove our main results,
which are as follows.
\begin{theorem} \label{thm3.2}
Assume that $u_0(x)$ and $v_0(x)$ are non-negative functions and that
$(u_0,v_0)$ belongs to $(W^{1,\theta}{(\Omega)})^2$
for some $\theta>n$, $\chi>0$, $f(u)$ satisfies \eqref{e1.2}.
Then problem \eqref{e1.1} possesses a unique global classical solution
$(u,v)$ for which both $u$ and $v$ are non-negative and uniformly bounded
in $\Omega\times(0,\infty)$ provided that
$$
0